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Calculus II — Worksheet 7 Page 1 of 2
Integration Techniques
Complete all problems. Show your work.
01
Evaluate using integration by parts.
$$\int x^2 e^{3x}\, dx$$
02
Evaluate using partial fraction decomposition.
$$\int \frac{5x^2 - 3x + 2}{x^3 - x}\, dx$$
03
Evaluate using trigonometric substitution.
$$\int \frac{\sqrt{x^2 - 9}}{x}\, dx$$
04
Determine whether the improper integral converges or diverges.
$$\int_1^{\infty} \frac{\ln x}{x^2}\, dx$$
Calculus II · MTH 162 Prof. Harrison

Solution

Evaluate $$\int x^2 e^{3x}\, dx$$ using integration by parts.

Step 1: Set up IBP with $u = x^2$ and $dv = e^{3x}\,dx$.

Then $du = 2x\,dx$ and $v = \tfrac{1}{3}e^{3x}$.

Step 2: Apply the formula $\int u\,dv = uv - \int v\,du$:

$$\int x^2 e^{3x}\,dx = \frac{x^2 e^{3x}}{3} - \frac{2}{3}\int x e^{3x}\,dx$$

Step 3: Apply IBP again for $\int x e^{3x}\,dx$:

$$\int x e^{3x}\,dx = \frac{x e^{3x}}{3} - \frac{e^{3x}}{9}$$

Step 4: Combine and simplify:

$$\boxed{\int x^2 e^{3x}\,dx = \frac{e^{3x}}{27}\left(9x^2 - 6x + 2\right) + C}$$

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Solution

Using Newton's second law, $F = ma$:

Given $F = 25\text{ N}$ and $m = 5\text{ kg}$:

$$a = \frac{F}{m} = \frac{25}{5} = 5\ \text{m/s}^2$$

The acceleration is 5 m/s² in the direction of the applied force.

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Integration by Parts
When you have a product of two functions where one simplifies when differentiated, use IBP:
$\int u\,dv = uv - \int v\,du$
Tip: Use LIATE to pick $u$: Logs → Inverse trig → Algebraic → Trig → Exponential
Try this
Evaluate $\int x\,\ln x\,dx$ using integration by parts.
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Good setup! You correctly identified $u$ and $dv$. Watch the sign when substituting back — that's where the error crept in.
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Solving 4 questions…
Page 1
Q1 $\frac{e^{3x}}{27}(9x^2 - 6x + 2) + C$
▸
1 Let $u=x^2$, $dv=e^{3x}dx$
2 Apply IBP twice to reduce the power of $x$
3 Factor out $e^{3x}/27$
Q2 $2\ln|x| - \frac{1}{2}\ln|x-1| + \frac{3}{2}\ln|x+1| + C$
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Q3 $\sqrt{x^2-9} - 3\sec^{-1}(x/3) + C$
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